Course in Probability Theory

8 .2 BASIC NOTIONS 1 273Instead of requiring a to be defined on all 0 but possibly taking the valueoc, we may suppose it to be defined on a set A in Note that a strictlypositive integer n is an optional r.v . The concepts of pre-a and post-a fieldsreduce in this case to the previous ~„ and ~~, ; .A vital example of optional r.v . is that of the first entrance time into agiven Borel set A :00min{n E N : Z n (w) E A} on U {w : Z n (w) E A) ;(5) aA(w) = n=1+00 elsewhere .To see that this is optional, we need only observe that for each n E N :{w : aA(w) = n } _ {w : Z j (w) E A`, 1 < j < n - 1 ; Zn (w) E A}which clearly belongs to ?,T, ; similarly for n = oo .Concepts connected with optionality have everyday counterparts, implicitin phrases such as "within thirty days of the accident (should it occur)" . Historically,they arose from "gambling systems", in which the gambler choosesopportune times to enter his bets according to previous observations, experiments,or whatnot . In this interpretation, a + 1 is the time chosen to gambleand is determined by events strictly prior to it. Note that, along with a, a + 1is also an optional r .v ., but the converse is false .So far the notions are valid for an arbitrary process on an arbitrary triple .We now return to a stationary independent process on the specified triple andextend the notion of "shift" to an "a-shift" as follows : ra is a mapping on{a < oo} such that(6) racy = r n w on {w : a(co) = n} .Thus the post-a process is just the process IX,, (taco), n E N} . Recalling thatX n is a mapping on S2, we may also write(7) Xa+n (w) = Xn (r a (0) = (Xn ° 'r ' ) (0) )and regard X„ ° ra, n E N, as the r .v.'s of the new process . The inverse setmapping (ra) -1 , to be written more simply as r-a, is defined as usual :r-'A = {w : raw E A} .Let us now prove the fundamental theorem about "stopping" a stationaryindependent process .Theorem 8 .2 .2 . For a stationary independent process and an almost everywherefinite optional r .v . a relative to it, the pre-a and post-a fields are independent. Furthermore the post-a process is a stationary independent processwith the same common distribution as the original one .